I decided to have a look at the Ultimate Source of All Necessary Information, i.e. Wikipedia.
I thought that the first line of the page for what schizophrenia is was absolutely hilarious.

"Schizophrenia ... is a mental disorder often characterized by abnormal social behavior and failure to recognize what is real."
"OFTEN" chracaterised by "ABNORMAL" social behaviour and failure to recognise what is "REAL". Ridiculous!
But anyway... it mentions that the ICD-10 proposes the category of "Simple schizophrenia: Insidious and progressive development of prominent negative symptoms with no history of psychotic episodes". Supposedly very rare, but it exists.
Now for the DSM-IV criteria of schizophrenia:
"DSM-IV-TR: Diagnostic criteria for schizophrenia: A. Characteristic symptoms:
Two (or more) of the following, each present for a significant portion of time during a 1-month period (or less if successfully treated): (1) delusions (2) hallucinations (3) disorganized speech (e.g., frequent derailment or incoherence (4) grossly disorganized or catatonic behaviour
(5) negative symptoms, i.e., affective flattening, alogia (poverty of speech), or avolition (lack of motivation) Note: Only one Criterion A symptom is required if delusions are bizarre or hallucinations consist of a voice keeping up a running commentary on the person's behavior or thoughts, or two or more voices conversing with each other. B. Social/occupational dysfunction: For a significant portion of the time since the onset of the disturbance, one or more major areas of functioning such as work, interpersonal relations, or self-care are markedly below the level achieved prior to the onset (or when the onset is in childhood or adolescence, failure to achieve expected level of interpersonal, academic, or occupational achievement). C. Duration: Continuous signs of the disturbance persist for at least 6 months. This 6-month period must include at least 1 month of symptoms (or less if successfully treated) that meet Criterion A (i.e., active-phase symptoms) and may include periods of prodromal (symptomatic of the onset) or residual symptoms. During these prodromal or residual periods, the signs of the disturbance may be manifested by only negative symptoms or two or more symptoms listed in Criterion A present in an attenuated form (e.g., odd beliefs, unusual perceptual experiences). D. Schizoaffective and Mood Disorder exclusion..."
Negative symptoms are only one of a list of 5 general symptoms, of which only 2 (or even 1) are required, and it says nothing about negative symptoms needing to be one of them.
So I gather from this that there are many supposed schizophrenics who don't have negative symptoms, though not many who don't have positive symptoms.
Note that you can have positive AND negative symptoms for any intensity or length of time and not be schizophrenic, so long as you can work/study, take care of yourself, and have good interpersonal relations.